Transcript Document 7343669

Portfolio Loss Distribution
Risky assets in loan portfolio
•
•
highly illiquid assets
“hold-to-maturity” in the bank’s balance sheet
Outstandings
The portion of the bank asset that has already been extended to
borrowers.
Commitment
A commitment is an amount the bank has committed to lend. Should
the borrower encounter financial difficulties, it would draw on this
committed line of credit.
Adjusted exposure and
expected loss
Let a be the amount of drawn down or usage given default.
Outstanding + a  commitment,
Risky
Asset value at
later time H, VH
(1-a) commitment,
Riskless
Adjusted exposure is the risky part of VH.
Expected loss = adjusted exposure  loss given default
 probability of default
* Normally, practitioners treat the uncertain draw-down rate as a
known function of the obligor’s end-of-horizon credit class rating.
Example
calculation of expected loss
Commitment
Outstanding
Internal risk rating
Maturity
Type
Unused drawn-down on default
(for internal rating = 3)
Adjusted exposure on default
EDF for internal rating = 3
Loss given default for non-secured asset
Expected loss
$10,000,000
$3,000,000
3
1 year
Non-secured
65%
$8,250,000
0.15%
50%
$6,188
Unexpected loss
Unexpected loss is the estimated volatility of the potential loss in value
of the asset around its expected loss.
2
2
UL  AE  EDF  LGD
 LGD 2   EDF
where
2
 EDF
 EDF  (1 - EDF).
Assumptions
* The random risk factors contributing to an obligor’s default (resulting
in EDF) are statistically independent of the severity of loss (as given by
LGD).
* The default process is two-state event.
Example on unexpected
loss calculation
Adjusted exposure
$8,250,000
EDF
0.15%
EDF
3.87%
LGD
50%
LGD
25%
Unexpected loss
$178,511
* The calculated unexpected loss is 2.16% of the adjusted exposure,
while the expected loss is only 0.075%
Comparison between expected
loss and unexpected loss
* The higher the recovery rate (lower LGD), the lower is the percentage
loss for both EL and UL.
* EL increases linearly with decreasing credit quality (with increasing
EDF)
* UL increases much faster than EL with increasing EDF.
Percentage loss per
unit of adjusted loss
10%
UL
EL
5%
EDF
10%
Assets with varying terms
of maturity
*
The longer the term to maturity, the greater the variation in asset
value due to changes in credit quality.
*
The two-state default process paradigm inherently ignores the
credit losses associated with defaults that occur beyond the
analysis horizon.
*
To mitigate some of the maturity effect, banks commonly adjust
a risky asset’s internal credit class rating in accordance with its
terms to maturity.
Portfolio expected loss
EL p   EL i   AE i  LGD i  EDFi
i
i
where ELp is the expected loss for the portfolio,
AEi is the risky portion of the terminal value of the ith asset
to which the bank is exposed in the event of default.
We may write
EL p
ELi
  wi
AE p
AEi
i
where the weights refer to
AE i
AE i
wi 

.
 AE i AE p
i
ELp
AE p
i
1
2
3
AEi
$10 M
$4 M
$6 M
EL
AE



AE
AE
 AE  $20M
i
ELp
AE p
i
i
p
p
wi
0.5
0.2
0.3
w 1
ELi
ELi
 wi
AEi
AEi
ELi
$1
$0.5
$0.6
ELi/AEi
0.1
0.125
0.1
i
 0.5  0.1  0.2  0.125  0.3  0.1  0.105
Portfolio unexpected loss
portfolio unexpected loss  UL p 
 
i
ij
wi w j UL i UL j
j
where
2
2
2
UL i  AE i  EDFi   LGD

L
GD


i
EDFi
i
and ij is the correlation of default between asset i and asset j. Due to
diversification effect, we expect
UL p   UL i .
i
Risk contribution
The risk contribution of a risky asset i to the portfolio unexpected loss
is defined to be the incremental risk that the exposure of a single asset
contributes to the portfolio’s total risk.
RC i  UL i
UL p
UL i
and it can be shown that
RC i 
UL i  UL j  ij
j
UL p
.
Undiversifiable risk
The risk contribution is a measure of the undiversifiable risk of an
asset in the portfolio – the amount of credit risk which cannot be
diversified away by placing the asset in the portfolio.
UL p   RC i
i
To incorporate industry correlation, using i  industry a and
j  industry b
ULi
RC i 
UL p


 
 ULia (1 - aa )     ULk ab .
b a  kb

 
Calculation of EL, UL and RC
for a two-asset portfolio

ELp
ULp
default correlation between the two exposures
portfolio expected loss
ELp = EL1 + EL2
portfolio unexpected loss
UL p  UL21  UL22  2 UL1UL 2
RC1
risk contribution from Exposure 1
RC2
risk contribution from Exposure 2
RC1  UL1 ( UL1  UL 2 ) / UL p
RC 2  UL 2 ( UL 2  UL1 ) / UL p
ULp = RC1 + RC2
ULp << UL1 + UL2
Fitting of loss distribution
The two statistical measures about the credit portfolio are
1. portfolio expected loss;
2. portfolio unexpected loss.
At the simplest level, the beta distribution may be chosen to fit the
portfolio loss distribution.
Reservation
A beta distribution with only two degrees of freedom is
perhaps insufficient to give an adequate description of
the tail events in the loss distribution.
Beta distribution
The density function of a beta distribution is
 (a(a)  (bb) ) xa -1 (1 - x) b -1 , 0  x  1

F ( x, a , b )  

0
otherwise

a  0, b  0
ab
Mean  
and variance  
.
2
(a  b ) (a  b  1)
ab
a
2
f(x, a, b)
x
1
Economic Capital
If XT is the random variable for loss and z is the percentage probability
(confidence level), what is the quantity v of minimum economic capital
EC needed to protect the bank from insolvency at the time horizon T
such that
Pr[ X T  v]  z.
Here, z is the desired debt rating of the bank, say, 99.97% for an AA
rating.
frequency of loss
ULp
XT
ELp
EC
Capital multiplier
Given a desired level of z, what is EC such that
Pr[ X T - EL p  EC ]  z.
Let CM (capital multiplier) be defined by
EC  CM  UL p
then
 X T - EL p

Pr 
 CM   z.
 UL p

Monte Carol simulation
of loss distribution of a portfolio
1. Estimate default and
losses
Assign risk ratings to loss
facilities and
determine their default
probability
+ Assign LGD
and LGD
2. Estimate asset correlation
between obligors
Determine pairwise
asset correlation
whenever possible
OR
Assign obligors to industry
groupings, then determine
industry pair correlation
3.
Generate random loss
given default
Determine stochastic
loss given default
4.
+
Generate correlated
default events
Correlated
Decomposition
default
+ of covariance
matrix
events
+
Simulate default point
5. Loss calculation
Calculate facility
loss for each
scenario and obtain
portfolio loss
6.
Loss distribution
Construct simulated
portfolio loss
distribution
Generation of correlated
default events
1. Generate a set of random numbers drawn from a standard normal
distribution.
2. Perform a decomposition (Cholesky, SVD or eigenvalue) on the
asset correlation matrix to transform the independent set of random
numbers (stored in the vector e ) into a set of correlated asset
values (stored in the vector e  ). Here, the transformation matrix is
M, where
e = M e .
The covariance matrix  and M are related by
M T M  .
Calculation of the
default point
The default point threshold, DP, of the ith obligor can be defined as
DP = N-1(EDFi, 0, 1). The criterion of default for the ith obligor is
if
e 'i  DPi
no default if
e 'i  DPi .
default
Generate loss given default
The LGD is a stochastic variable with an unknown distribution.
A typical example may be
Recovery rate (%)
LGD (%)
LGD (%)
secured
65
35
21
unsecured
50
50
28
s
LGD i  LGD s  f i   LGD
where fi is drawn from a uniform distribution whose range is selected
so that the resulting LGD has a standard deviation that is consistent
with historical observation.
Calculation of loss
Summing all the simulated losses from one single scenario
Loss 

Adjusted exposure i  LGD
Obligors
in default
Simulated loss distribution
The simulated loss distribution is obtained by repeating the above
process sufficiently number of times.
Features of portfolio risk
•
The variability of default risk within a portfolio is substantial.
•
The correlation between default risks is generally low.
• The default risk itself is dynamic and subject to large fluctuations.
•
Default risks can be effectively managed through diversification.
•
Within a well-diversified portfolio, the loss behavior is
characterized by lower than expected default credit losses for much
of the time, but very large losses which are incurred infrequently.